Periodicity and roots of transfinite strings

Transfinite Equations in Transfinite Strings

The integer transfinite diameter of intervals and totally real algebraic integers

In this paper we build on some recent work of Amoroso, and Borwein and Erdélyi to derive upper and lower estimates for the integer transfinite diameter of small intervals + 03B4], is a fixed rational and 03B4 ~ 0. We also study functions g-, g, g+ associated with transfinite diameters of Farey intervals. Then we consider certain polynomials, which we call critical polynomials, associated to a g...

Trace of totally positive algebraic integers and integer transfinite diameter

Explicit auxiliary functions can be used in the “Schur-SiegelSmyth trace problem”. In the previous works, these functions were constructed only with polynomials having all their roots positive. Here, we use several polynomials with complex roots, which are found with Wu’s algorithm, and we improve the known lower bounds for the absolute trace of totally positive algebraic integers. This improve...

The Theory of Rational Relations on Transfinite Strings

In order to explain the purpose of this paper, we recall briefly how the theory of “rationality” developed in the last fourty years. The theory of sets of finite strings recognized by finite automata, also known as regular or rational sets was developed in the fifties. It rapidly extended in two directions. Indeed, by solving the decidability problem of the second order monadic theory of one su...

Elementary Theory of Ordinals with Addition and Left Translation by omega

After Büchi it has become very natural to interprete formulae of certain logical theories as finite automata, i.e., as recognizing devices. This recognition aspect though, was neglected by the inventor of the concept and the study of the families of linear structures that could be accepted in the language theory sense of the term, was carried out by other authors. The most popular field of appl...